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14
BASIC CHARACTERS OF THE UNITRIANGULAR GROUP (FOR ARBITRARY PRIMES)
, 2002
"... Let Un(q) denote the (upper) unitriangular group of degree n over the finite field Fq with q elements. In this paper we consider the basic (complex) characters of Un(q) and we prove that every irreducible (complex) character of Un(q) is a constituent of a unique basic character. This result extends ..."
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Let Un(q) denote the (upper) unitriangular group of degree n over the finite field Fq with q elements. In this paper we consider the basic (complex) characters of Un(q) and we prove that every irreducible (complex) character of Un(q) is a constituent of a unique basic character. This result extends a previous result which was proved by the author under the assumption p ≥ n, wherepisthe characteristic of the field Fq.
Character degree graphs that are complete graphs
 Proc. AMS 135
, 2007
"... Abstract. Let G be a finite group, and write cd(G) for the set of degrees of irreducible characters of G. We define Γ(G) to be the graph whose vertex set is cd(G) −{1}, and there is an edge between a and b if (a, b)> 1. We prove that if Γ(G) is a complete graph, then G is a solvable group. 1. ..."
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Cited by 2 (2 self)
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Abstract. Let G be a finite group, and write cd(G) for the set of degrees of irreducible characters of G. We define Γ(G) to be the graph whose vertex set is cd(G) −{1}, and there is an edge between a and b if (a, b)> 1. We prove that if Γ(G) is a complete graph, then G is a solvable group. 1.
Nonsolvable Groups with No Prime Dividing Three Character Degrees
, 2010
"... Throughout this note, G will be a finite group, Irr(G) will be the set of irreducible characters of G, and cd(G) will be the set of character degrees of G. We consider groups where no prime divides at least three degrees in cd(G). Benjamin studied this question for solvable groups in [1]. She proved ..."
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Cited by 2 (2 self)
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Throughout this note, G will be a finite group, Irr(G) will be the set of irreducible characters of G, and cd(G) will be the set of character degrees of G. We consider groups where no prime divides at least three degrees in cd(G). Benjamin studied this question for solvable groups in [1]. She proved that solvable groups with this property satisfy cd(G)  � 6. She also presented examples to show
D.: Frobenius polytopes
 Preprint, http://www.reed.edu/ davidp/homepage/mypapers/frob.pdf (2004) BAUMEISTER, HAASE, NILL, AND PAFFENHOLZ
"... Abstract. A real representation of a finite group naturally determines a polytope, generalizing the wellknown Birkhoff polytope. This paper determines the structure of the polytope corresponding to the natural permutation representation of a general Frobenius group. 1. ..."
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Abstract. A real representation of a finite group naturally determines a polytope, generalizing the wellknown Birkhoff polytope. This paper determines the structure of the polytope corresponding to the natural permutation representation of a general Frobenius group. 1.
HECKE ALGEBRAS FOR THE BASIC CHARACTERS OF THE UNITRIANGULAR GROUP
, 2003
"... Let Un(q) denote the unitriangular group of degree n over the finite field with q elements. In a previous paper we obtained a decomposition of the regular character of Un(q) as an orthogonal sum of basic characters. In this paper, we study the irreducible constituents of an arbitrary basic character ..."
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Cited by 1 (1 self)
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Let Un(q) denote the unitriangular group of degree n over the finite field with q elements. In a previous paper we obtained a decomposition of the regular character of Un(q) as an orthogonal sum of basic characters. In this paper, we study the irreducible constituents of an arbitrary basic character ξ�(ϕ) ofUn(q). We prove that ξ�(ϕ) is induced from a linear character of an algebra subgroup of Un(q), and we use the Hecke algebra associated with this linear character to describe the irreducible constituents of ξ�(ϕ) as characters induced from an algebra subgroup of Un(q). Finally, we identify a special irreducible constituent of ξ�(ϕ), which is also induced from a linear character of an algebra subgroup. In particular, we extend a previous result (proved under the assumption p ≥ n where p is the characteristic of the field) that gives a necessary and sufficient condition for ξ�(ϕ) to have a unique irreducible constituent.
NORMAL SUBGROUPS OF ODD ORDER MONOMIAL P A Q BGROUPS BY
"... ii A finite group G is called monomial if every irreducible character of G is induced from a linear character of some subgroup of G. One of the main questions regarding monomial groups is whether or not a normal subgroup N of a monomial group G is itself monomial. In the case that G is a group of ev ..."
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ii A finite group G is called monomial if every irreducible character of G is induced from a linear character of some subgroup of G. One of the main questions regarding monomial groups is whether or not a normal subgroup N of a monomial group G is itself monomial. In the case that G is a group of even order, it has been proved (Dade, van der Waall) that N need not be monomial. Here we show that, if G is a monomial group of order p a q b, where p and q are distinct odd primes, then any normal subgroup N of G is also monomial. iii to my parents Anna and Giannis, and to my teacher, Everett C. Dade, without whom none of this would have been done, and everything would have been written faster. iv Acknowledgments First and foremost I would like to thank my advisor Prof. Everett C. Dade not only for his endless patience, his continued support and encouragement, his creativity and his humor, but also for all those afternoon meetings in his office, where I saw how mathematics can become pure art. Thank you. This would have been only a dream without your help. I would also like to thank the members of my committee that went through the trouble of reading this thesis, and Prof. Marty Isaacs for his sincere interest in this work. I also thank my first math teacher Manolis Mpelivanis, Prof. George Akrivis for his faith in me, and Prof. Giannis Antoniadis for introducing me to Representation Theory. Special thanks go to my parents Anna and Giannis and my sister Marianthi for always being there, and to Michalis for the verses at Daily Grind and the “sweet and sour ” evenings. I’m grateful to all my friends that made my stay in Urbana memorable, especially Sasa (για oλα), George (για κατι “ɛνδιαφɛρoν και διασκɛδαστικo”), Michalino (για τιs κoυβɛντɛs στo
PRIMITIVE CENTRAL IDEMPOTENTS OF FINITE GROUP RINGS OF SYMMETRIC GROUPS
"... Abstract. Let p be a prime. We denote by Sn the symmetric group of degree n, byAn the alternating group of degree n and by Fp the field with p elements. An important concept of modular representation theory of a finite group G is the notion of a block. The blocks are in onetoone correspondence wit ..."
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Abstract. Let p be a prime. We denote by Sn the symmetric group of degree n, byAn the alternating group of degree n and by Fp the field with p elements. An important concept of modular representation theory of a finite group G is the notion of a block. The blocks are in onetoone correspondence with block idempotents, which are the primitive central idempotents of the group ring FqG, whereqis a prime power. Here, we describe a new method to compute the primitive central idempotents of FqG for arbitrary prime powers q and arbitrary finite groups G. For the group rings FpSn of the symmetric group, we show how to derive the primitive central idempotents of FpSn−p from the idempotents of FpSn. Improving the theorem of Osima for symmetric groups we exhibit a new subalgebra of FpSn which contains the primitive central idempotents. The described results are most efficient for p =2. Inan appendix we display all primitive central idempotents of F2Sn and F4An for n ≤ 50 which we computed by this method. Introduction and notation
IRREDUCIBLE CHARACTERS OF FINITE ALGEBRA GROUPS
, 1998
"... Let p be a prime number, let q = p e (e ≥ 1) be a power of p and let Fq denote the finite field with q elements. Let A be a finite dimensional Fqalgebra. (Throughout the paper, all algebras are supposed to have an identity element). Let J = J(A) be the Jacobson radical of A and let ..."
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Let p be a prime number, let q = p e (e ≥ 1) be a power of p and let Fq denote the finite field with q elements. Let A be a finite dimensional Fqalgebra. (Throughout the paper, all algebras are supposed to have an identity element). Let J = J(A) be the Jacobson radical of A and let